Rings arising from finite tight hypergroups (Q2216556)

The notion of a hypergroup (in the sense of Marty) provides a far reaching and meaningful generalization of the concept of a group. Specific classes of hypergroups have given rise to challenging questions and interesting connections to geometric and group theoretic topics. The present article is a continuation of a study of residually thin hypergroups which was initiated in [\textit{C. French} and \textit{P.-H. Zieschang}, J. Algebra 551, 93--118 (2020; Zbl 1487.20031)]. One of the main results of [loc. cit.] says that finite tight hypergroups, that is hypergroups all elements \(h\) of which satisfy \(hh^*h = \{h\}\), are residually thin. In this paper, the authors take advantage of this result in order to construct, for each finite tight hypergroup satisfying a mild extra condition, an associative ring. The rings which they found generalize the construction of scheme rings for a certain class of association schemes, including those which correspond to finite groups (group rings). Remark: The hypergroup that the authors study in this paper (a special case of hypergroups) is usually called polygroup, see [the reviewer, Polygroup theory and related systems. Hackensack, NJ: World Scientific (2013; Zbl 1266.20083)].